非正交融合框架与融合框架算子的稀疏性
Non-orthogonal fusion frames and the sparsity of fusion frame operators
摘要：
融合框架已经成为实现分布式系统的主要工具。融合框架在分布式系统中应用的有效性体现在最终融合过程的效率上。这又反映在融合帧算子$S_{\cW}$的反演效率中，其又严重依赖于$S_{\cW}$的稀疏性。我们将通过引入{ \it非正交融合框架}的概念来证明融合框架算子的稀疏性自然存在。我们表明，对于一个融合框架$\{W_i,v_i\}_{i\in I}$，如果$\text{dim}(W_i)=k_i$，那么矩阵的非正交融合框架运营商$\cSw$在其相应的位置最多有一个$k_i\times k_i$块矩阵。我们提供了新的融合框架运营商$\cSw$是对角线和/或多个单位的充要条件。一系列其他关键问题也得到了解决。还检查了{ \it多个融合帧}的方案，其对应的融合帧算子变为对角算子。
英文摘要：
Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion frame applications in distributed systems is reflected in the efficiency of the end fusion process. This in turn is reflected in the efficiency of the inversion of the fusion frame operator $S_{\cW}$, which in turn is heavily dependent on the sparsity of $S_{\cW}$. We will show that sparsity of the fusion frame operator naturally exists by introducing a notion of {\it non-orthogonal fusion frames}. We show that for a fusion frame $\{W_i,v_i\}_{i\in I}$, if $\text{dim}(W_i)=k_i$, then the matrix of the non-orthogonal fusion frame operator $\cSw$ has in its corresponding location at most a $k_i\times k_i$ block matrix. We provide necessary and sufficient conditions for which the new fusion frame operator $\cSw$ is diagonal and/or a multiple of an identity. A set of other critical questions are also addressed. A scheme of {\it multiple fusion frames} whose corresponding fusion frame operator becomes an diagonal operator is also examined.
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